Question 1

what is the evaluation and reaction to the article

Question 2

implication o the research for special education in the regular classroom

Mathematical learning is essential for students' future successbecauseitencouragesinquisitiveness,developscritical thinking, and fostersproblem-solving skills (Clementset al., 2004). Mathematics also provides young people withthe fundamental knowledge needed to access science, engineering, and technology (Vordermanet al., 2011). Theseindustries are expected to provide young people withincreasing job opportunities in the future (Sargent, 2017). However,asimportantasmathematicsisforfuturesuccess, many students perceive mathematics to be a difficult subject to master. Approximately, 10% to 20% of school-age students are reported to have significant difficulties withmathematics (Geary, 2011). Furthermore, depending on thecriteria used for identification and the cut points used fordetermination, between 3% and 9% of students are classified as having a learning disability (LD) in mathematics (Shalev et al., 2000; Shalev & von Aster,2008).

Many students with LDs show persistent, severedeficits in general and numerical cognition (Berch & Mazzocco, 2007; Geary et al., 2012). Specifically, students with LDoften have difficulty in representing numerical magnitude, retrieving basic arithmetic facts, and processing information(Berch&Mazzocco,2007;Geary,2011).Inadditiontoa lack of numeric skills, the students also show delays inworking memory, long-term memory, and organizational skills (Geary et al., 2012). Deficits in languagecomprehension, attentive behavior, nonverbal problem-solving, andlistening span also negatively affect these students' mathematical problem-solving skills (Fuchs et al., 2010).

Such mathematical and cognitive difficulties impose significant barriers to appropriate mathematics growth atevery grade level. Research on longitudinal mathematics outcomesindicatesthatstudentsidentifiedwithLDachieve significantly less growth in mathematics than students without LD (Judge & Watson, 2011; Morgan et al., 2016). The gap in mathematics achievement between students with and without LD becomes wider with every year of schooling (Stevens & Schulte, 2017). Many students withLD move to higher grades without establishing a fundamental knowledge of mathematics, and their struggle in learning mathematics intensifies by the year. Therefore, itis important to examine previous research in identifying effective instructional strategies for teaching mathematics to students withLD.

Mathematics RepresentationsforStudents WithLD

Oneapproachtoimprovingthemathematicsachievementofstudents with LD is the use of mathematics representations(Jitendra et al., 2016). A mathematical representationis

defined as "a sign, a configuration of signs, character, orobject that stands for something (symbolizes, depicts,encodes, or represents) other than itself" (Goldin & Shteingold,2001,p.3).Mathematicsrepresentationsincludeinternal representations (i.e., learners' mental configura-tions) and external representations (i.e., physical or observ-able configurations such as graphs, diagrams, pictures,tables,formulas,symbols,words,gestures,videos,concrete andvirtualmanipulatives,tabularforms,andsounds;Goldin& Shteingold,2001).

School children can better understand mathematics bydeveloping knowledge of how to represent mathematics features and quantities (Powell & Fluhler, 2020). Students can participate in three different modes of representations: "enactive"(physicalmanipulatives),"iconic"(pictorialrep-resentation), and "symbolic" (abstract mathematical sym-bols) representations (Bruner, 1964, p. 2). The National Mathematics Advisory Panel (2008) recommended the useof visual and multiple representations as an effective strat-egy for teaching mathematical concepts. Furthermore, the National Council of Teachers of Mathematics's (NCTM,2000) Principles and Standards for School Mathematics includes a standard stating that mathematics instruction should provide every K-12 student with opportunities toutilize multiple representations for problem-solving. The What WorksClearinghouse's practice guide also suggeststhatinterventionsshouldincludetheuseofvisualrepresentations of problem structures or mathematical concepts forstudents with mathematics difficulties, including students with LD (Gersten et al., 2009; Woodward et al.,2012).

The use of multiple representations supports learners as they connect different mathematical features and properties, thereby assisting them in understanding mathematical concepts through analysis and interaction with various representations (NCTM, 2014). For instance, students can restate a mathematical concept using their own words (a verbal representation), draw diagrams or graphs (a graphical or pictorial representation), manipulate physical objects (a concrete representation), or write an equation (a sym- bolic representation). In schools, the use of mathematics manipulatives (i.e., representations of physical objects) is a common approach in applying multiple representations (Bouck & Park, 2018).

Research syntheses have demonstrated the positive effects of interventions using manipulatives for students with LD across mathematical domains and grade levels (Agrawal & Morin, 2016; Lafay et al., 2019). As aninstructional strategy incorporating mathematics manipulatives, a graduated instructional framework has been usedfor students with LD (Bouck, Satsangi, & Park, 2018). This graduated instructional framework (Concrete- Representational-Abstract framework [CRA]) is a strategy that accords with mathematics representations using manipulatives. In the concrete phase, students learn

mathematical concepts and solve mathematics problems by using physical manipulatives. Using physical manipu- latives, students can model a mathematical concept or rep-resent the structure of a problem while in the process ofsolving the problem. Students then move to semi-concrete representations (e.g., visual drawings) in the representa- tional phase, and finally, in the abstract phase, they under- stand the mathematics concepts and solve mathematics problems using symbolic representations without concrete or semi-concrete representations (Bouck, Satsangi, & Park, 2018). The graduated instructional framework hasbeen recognized as an evidence-based approach for stu- dents with LD (Bouck, Satsangi, & Park,2018).

ResearchontheUseofVirtualManipulatives

As a technology-based tool for mathematics learning, a vir-tual manipulative is defined as "an interactive, technology- enabled visual representation of a dynamic mathematical object, including all of the programmable features thatallow it to be manipulated, that presents opportunities forconstructing mathematical knowledge" (Moyer-Packenham& Bolyard, 2016, p. 13). Virtual manipulatives function similar to physical manipulatives, but virtual manipulativeshave developed side by side with the advancement of com-putertechnology.Theintentbehindvirtualmanipulativesisto represent mathematical ideas and to provide students with opportunities to manipulate dynamic objects on ascreen. With virtual manipulatives, students can use visual cues and prompts on a screen to visualize the structure of amathematics problem and solve the problem (Moyer-Packenham et al., 2012; Moyer-Packenham & Suh, 2012). In mathematics intervention, virtual manipulatives providea wide variety of options for students' learning, thus aidingeducators in meeting each individual's academic needs (Satsangi et al., 2016). Interactive visual models can pro-vide scaffolded mathematical content, increase opportuni- ties to practice mathematical problems through a virtual screen, and enhance students' engagement in mathematics learning activities (Shin et al., 2017). In addition, virtual manipulatives can provide a new means of representation, which aligns with a primary principle of the Universal Design for Learning (Bouck & Flanagan, 2009).

Compared with physical manipulatives, virtual manipu- latives can be quickly used through a screen, provide aninfinite number of manipulative objects, and allowflexibil- ity in the size and color of the manipulatives (Bouck &Flanagan, 2009; Moyer-Packenham & Bolyard, 2016). Furthermore,computersortabletscommonlyprovideassis- tant features (e.g., assistive touch, adaptive mouse, or key- board), which facilitate the learning activities of students who need additional support (Ok, 2018). The inherent sup- portsorconstraintswithinvirtualmanipulativesmayalso

provide the possibility of decreasing students' cognitive loads (Suh & Moyer-Packenham, 2008). With its increased flexibility and accessibility, virtual manipulatives can pro- vide representational supports in inclusive school settings without stigmatization (Moyer-Packenham et al., 2012; Reimer&Moyer-Packenham,2005).Anotherdistinctionofvirtualmanipulativesistheirflexibleuseinvarioustechno- logical environments (Moyer-Packenham & Bolyard, 2016).Themostcommontypesofvirtualmanipulativesaredesignedas(a)single-representation(i.e.,pictorialorvisual representation without any additional information), (b)multi-representation (i.e., two or more forms of representa- tions), (c) tutorial (i.e., guiding and tutoring features fortheusers in addition to multi-representations), (d) gaming (i.e.,manipulating virtual manipulatives while playing a game),and (e) simulation (i.e., running a simulation of virtualmanipulatives that represents mathematics concepts).

Several studies have examined the effects of virtual manipulatives on students' academic abilities and the results show that the use of virtual manipulatives provides the greatest benefit to those students who struggle most in mathematics (Moyer-Packenham & Suh, 2012). By using virtual manipulatives to represent concepts, students who struggle with mathematics increase their likelihood of suc- cess in retrieving numerical information and representing mathematical concepts (Moyer-Packenham et al., 2012; Moyer-Packenham & Suh, 2012). Specifically, studies have shown that, after receiving interventions involving virtual manipulatives, low-achieving groups demonstrated statistically significant gains from the addition of virtual manipulatives (Moyer-Packenham & Suh, 2012). Virtual manipulatives also showed positive effects in inclusive classroom settings, supporting low-achieving learners by reshaping written words into visual objects (Reimer & Moyer-Packenham, 2005).

Toestablishaframeworkfortheuseofvirtualmanipula- tives for students with disabilities, Bouck et al. (2017) suggested the virtual-representational-abstract (VRA)framework, which was adapted from the CRA framework. Inamannersimilartotheevidence-basedCRAframework, theVRAframeworkemploysexplicitinstructionconsisting of three phases: virtual, representational, and abstract. Specifically, students attempt to solve mathematics prob-lems by manipulating digital objects during the virtual phase and then transferring those mathematics problems topictorial representations (e.g., drawings) during the repre- sentational phase. Finally, students solve the problemswithout using virtual manipulatives or drawing pictures(Bouck, Bassette, et al., 2017; Park et al., 2021).

Research syntheses have found that interventions using manipulatives show promising effects on the mathematics performance of students with disabilities, acknowledging the value of further research on virtual manipulatives. For example, Peltier et al. (2020) meta-analyzed 53 single-case

design studies examining the effects of manipulatives forstudentsatriskoridentifiedashavingdisabilities.Indicating a medium effect of manipulatives, Peltier et al. also ana-lyzed moderators such as manipulative type and found thatinterventions using virtual manipulatives showed greatereffects than those using physical manipulatives; however, the difference was not statistically significant. Bouck andPark (2018) also reviewed 36 studies regarding manipula- tives for students with disabilities and highlighted both theemergenceofvirtualmanipulativesandtheneedtoexamine the effects of virtual manipulatives as stand-alone tools infuture research. Focusing on students with LD, Lafay et al.(2019) synthesized 38 studies and noted a wide variation inthe types of manipulatives (e.g., physical, pictorial, andvir-tual) across all reviewed studies, but they found that sevenstudies that examined the effects of virtual manipulatives providednodetailsofhowthestudentsworkedwithdigital objects in their mathematics learning.

Purpose and Research Questions ofthe Current Study

AlthoughamajorityofstudentswithLD(72%ofthetotal numberofenrolledstudentswithLD)receiveeducational services in inclusive settings for 80% or moreof their school hours, many students with LD strugglewithengag-ingandpersistinginlearningmathematics(Snyderetal., 2019). Toincrease students' access to thegeneralmathe-maticscurriculumandtopromotethelearningofstudents withLD,educatorsneedtoensurethatstudentsactively participateintheirlearningprocesses.Beingtechnology- assistedmanipulatives,virtualmanipulativescanprovide representational supports in inclusive settings; and,insec-ondary and inclusive classrooms,technologicalsupports are considered more socially acceptable toolsthanphysical manipulatives (Bouck, Chamberlain, &Park,2017; Satsangi&Bouck,2015;Satsangietal.,2016).Furthermore, theuseofvirtualmanipulativeshasbecomeanimportant instructional online resource in teachingmathematicalcon-tent as distance learning formats increase (Serianni, 2014). Notingthebenefits,Moyer-PackenhamandWestenskow (2013)synthesizedtheeffectsofvirtualmanipulativeson students'mathematicslearning.Thissynthesisexamined66 primarystudiesandindicatedoverallmoderateeffectsof virtualmanipulativeswhencomparedwithotherinstruc-tionaltreatments(e.g.,classroominstructionusingatext-book). Moyer-Packenham and Westenskowsuggestedthe unique advantages of virtual manipulativesinmathematics learning, but the synthesis reported insufficientevidenceby student groups, indicating limited research on theeffectsof virtualmanipulativesforstudentswithLD.Althoughthe useofvirtualmanipulativeshasbeendeemedmoderately effectiveforpopulationswithoutdisabilitiesingeneral,itis necessary to determine whether the same effects arefound

among students with LD. Despite the emerging popularity of technology-mediated intervention for students with LD,evidencesupportingtheuseofvirtualmanipulativeshasnotbeen extensively examined (Bouck & Park, 2018; Satsangiet al., 2016). No systematic review has synthesized studies of interventions using virtual manipulatives to supportmathematics learning for students with LD. This research gap indicates a need for more investigation into effectivefeatures involving virtual manipulatives for students with LD.Therefore,thepurposeofthissynthesisistosystemati- cally review extant literature examining mathematics out-comes for K-12 students with LD when the mathematicalinterventions employ virtual manipulatives. The following research questions guided this synthesis:

Research Question 1 (RQ1):What are the features ofmathematics interventions using virtual manipulatives(e.g., devices, instructional practice, and virtual manipu- latives' features) to support students with LD?

Research Question 2 (RQ2):What are the effects of mathematics interventions using virtual manipulatives (e.g., immediate outcomes, maintenance, and generalization) on the mathematical achievement of students with LD?

Method

This synthesis represents a systematic review of the literature regarding the use of virtual manipulatives in teaching mathematics to K-12 students with LD. As a methodological procedure, the researcher followed the directions fromthe preferred reporting items for systematic reviews andmeta-analyses (PRISMA) statements. The PRISMA statement is a guideline for conducting and reportingsystematic reviews and meta-analyses. It consists of four phases: identification,screening,eligibility,andinclusion(Moheretal.,2009).

Search Strategies and Inclusion Criteria

In the identification phase, we identified articles in threeways. First, we conducted online searches of theAcademic

Search Complete (n= 257), APA PsycINFO (n= 200),Communication and Mass Media Complete (n=18),EducationSource(n=90),EducationResourcesInformationCenter(ERIC;n=77),andProQuest(n=473)database.In

the first line of the electronic search field, we targeted thefollowing words: virtualOR technologyOR computerORiPadOR tabletOR appOR web-basedOR onlineOR educationalsoftwareORinteractive.Inthesecondline,weusedmanipulativesOR number lineOR concreteOR object. Inthe third line, we used math* OR computationOR arithmetic- ticOR num* OR wordproblem OR solving. In the fourthline,weuseddisability*ORdisorderORdyscalculiaOR

special need. Journals and dissertations published fromJanuary 1980 to June 2020 in English were included,whichprocess returned a total of 1,115records; after removingduplicated studies, 930 records remained. Wedesignated1980 as the starting year because technology-mediated instruction saw a marked increase in implementationbegin-ninginthe1980sduetoconsiderabletechnologicalimprove-ments in the size and affordability of computers in the fieldofeducation(Woodward&Carnine,1993).Second,wecon-ducted an ancestral search of synthesis papers that met thetechnology-based mathematics interventions or the use ofmanipulatives for students with disabilities (Bouck & Park,2018; Bouck, Satsangi, & Park, 2018; Kiru et al., 2018; Lafayetal.,2019;Oketal.,2020;Ok&Kim,2017; Peltieret al., 2020; Stultz, 2017). Finally,we conducted a hand search of the following journals: Journal of LearningDisabilities, Learning Disabilities Quarterly, Learning Disabilities Research and Practice, Exceptional Children,and Remedial and Special Education. Wereviewed articlespublishedinthesejournalsfromJanuary1980toJune2020 and we identified no additional articles that met the inclusion criteria. The total initial search process returned 1,161 records.

In the process of screening and determining inclusion eligibility,theresearcherscreenedtheabstractsandtitlesofthe 1,161 initial search results and excluded 1,090 irrelevant records. The full texts of the 71 remaining recordswerescreenedaccordingtothefollowingfourinclusioncriteria. First, the studies needed to employ an experimental design showing quantitative data, thus allowing for the calculationofeffectsizes(ES).Groupdesignstudiesneededtopresent statistical information, employing randomized controlled trial or quasi-experimental designs. Single-subject design studies are needed to present data from both baseline and intervention phases. Second, students needed tomanipulate the virtual manipulatives for themselves. If only theteachers used virtual manipulatives and the students had noopportunity to manipulate objects during the intervention, the studies were excluded. Third, assessments of student performance are needed to show results for mathematics out-comes.Ifthestudiesincludedothervariablessuchasbehaviors (e.g., on-task, independence), the results needed topresent disaggregated data in mathematical outcomes. Fourth, the studies needed to include K-12 participantswith LD. When a study included participants with multiple disabilities, that study was included if the data were disaggregated for students with LD. According to the inclusion andexclusioncriteria,52recordswereexcludedforthefollowing reasons: (a) no quantitative data (n= 12), (b) no opportunities for students to manipulate virtual manipulatives(n=7),(c)nodataformathematicsoutcomes(n=5),and(d)nodisaggregateddataforstudentswithLD(n=

28). Asa result, theresearcher identified 19single-case

designstudies.Nogroupdesignstudymetourinclusion

criteria. Therefore, the final selection consisted of a total of19 single-case design studies from 16 journals and threedissertations (see online supplemental Figure S1). Before the primary researcher coded the 19 studies, a secondcoderchecked the interrater reliability of these selected studies. Using an Excel file, the second coder examined each studyto determine whether it met the four inclusion criteria (seeabove). Thereafter, we calculated interrater reliability bycounting all agreements divided by the total items ofagreements and disagreements and then multiplying by 100. Thetwocodersmet100%agreementontheinclusioncriteriaofall studies and decided to include them in the further, in-depth coding process.

Coding Procedures

Twocoders discussed each coding variable related toresearch questions and developed a coding tool using an Excelfilebasedonthereviewofpreviousliteratureregardingtheuseoftechnologyandvirtualmanipulativesinmathematicsinstruction(e.g.,Kiruetal.,2018;Moyer-Packenham & Bolyard, 2016; Ok et al., 2020). The coding protocolfocusedonthreecategories:(a)studyfeatures(e.g.,research design,participants,mathematicaldomains,andstudyquality), (b) features of mathematics interventions using virtualmanipulatives (e.g., instructional practice, interventionist, device,andvirtualmanipulatives'features),and(c)calculation of ES.

Studyfeatures.Weidentifiedresearchdesigns(e.g.,multi-ple-baseline design, alternating treatment design, or multi-ple-probe design) and coded the condition of each phase (i.e., baseline, intervention, maintenance, or generalization). Regarding participants, we coded grade and gender. Todetermine which mathematical domains were measuredin the intervention studies, we coded the Common Core StateStandardsforMathematics(CCSSM;NationalGovernors Association Center for Best Practices & Council ofChief State School Officers [NGA Center & CCSSO], 2010) of each study.Todetermine the methodological rigorof the included studies, we used quality indicators outlinedby the Council for Exceptional Children (CEC; Cook et al.,2014).Weusedthecode"1"whenthestudymetthedescriptionoftheCECqualityindicatorsandweusedthecode"0"when the quality indicator was not met. The maximumscore, for a study that met all criteria of the CEC indicators for single-case studies, was 22 points.

Features of mathematics interventions.In intervention features, we identified the types of instructional practicesincluded in each intervention (e.g., explicit instruction, agraduated instructional framework, and system of least prompts),interventionist(i.e.,researcher,teacher,ortutorialprogram), duration (i.e., minutes per session),frequency

(i.e.,timesperweek),thetotalnumberofsessions,andoverall session length (i.e., weeks). Regarding virtualmanipulatives' features, the device (i.e., computer or touch-based device), program and developer, and virtual manipulatives' type(e.g.,single-representation,multi-representations,tutorial, game, and simulation) were coded. Wealso identified the specific features by which students interacted with virtual manipulatives, inherent constraints, built-in supports, and any other differences from the physical manipulatives described in thestudies.

CalculationofES.Ifthestudiesincludedmultiplemeasurements (e.g., task completion, independence rate), we separately coded only the results of mathematics performance and identified the mathematical domains of the measurement. If the studies presented the results of interventions using both virtual and physical manipulatives throughalternating treatment design, we only coded the outcomes forthe use of virtual manipulatives.

Weidentified the total number of data points, the overall description of results for each participant, and the presence and efficacy of maintenance. Because most reviewed studies published their results using visual analysis, weused the WebPlotDigitizer program(https://automeris.io/WebPlotDigitizer/) to extract data points from the visual graphs (Rohatgi, 2020). Weuploaded a JPEG image ofeach graph into WebPlotDigitizer and exported the coordinates. The coordinates were organized into 19 Excel sheets for the 19 single-case studies, with each sheet including baseline, intervention, maintenance, and generalization data points for each participant. Weextracted a total of 695data points from the graph across baseline, intervention, maintenance, and generalization, thus categorizing the data into four phases: A (baseline data), B (intervention data),C (maintenance data), and D (generalizationdata).

Todetermine the effects of interventions, we calculatedTau-UES in this study.Tau-Uis a nonoverlap index thatquantifiesthetreatmentoutcomesofsingle-caseexperimental design data, involving corrections for baseline trends(Parkeretal.,2011).Tau-Ucanbeinterpretedas"thepercent of data that improve over time considering both phase non-overlap and Phase B trend, after control of Phase A trend"(Parker et al., 2011,p. 291). Wecalculated the number ofpairs (i.e., comparison of data points between phases percase) and the Tau-U,using a web-based application(http://www.singlecaseresearch.org;Vannestetal.,2016).Usingthedata points, we calculated Tau-Ufor baseline versus intervention (AB contrasts), Tau-Ufor intervention versusmaintenance (BC contrasts), and Tau-U for baseline versusgeneralization (AD contrasts). A Tau-Uof less than .20 isinterpreted as a small effect,a Tau-Ubetween .20 and .60isinterpretedasamoderateeffect,aTau-Ubetween.60and.80isinterpretedasalargeeffect,andaTau-Ugreaterthan.80is interpretedasaverylargeeffect(Vannest&Ninci,2015).

Interrater Reliability

The second coder independently checked all 19 studies andcompared each coding variable. Wecalculated interrater reliability by dividing all agreements by the total items of agreements and disagreements and then multiplying by 100.Thus, we achieved interrater reliabilities of 96.5% forstudyfeatures, 98.3% for features of mathematics interventionusing virtual manipulatives, and 100% for extracted single-case data. After discussing the disagreements, we reached 100% agreement on the final coding data.

Results

A total of 19 studies (16 peer-reviewed articles and threedissertations) published between 2014 and 2020 wereincluded in this synthesis. Weidentified the main featuresof mathematics interventions using virtual manipulativesand intervention effects. The 19 reviewed studies involved 35 participants from third to 12th grades who were identifiedashavingLD.Fromthe19studies,weextracteda total of695datapoints,92phasecontrasts(i.e.,43ABcontrasts, 30BCcontrasts,and19ADcontrasts),and2,645pairs(i.e.,1,732ABpairs,662BCpairs,and251ADpairs).Thus,wegenerated 43 Tau-Uto determine the immediate outcomes, 30 Tau-Ufor maintenance effects, and 19 Tau-Ufor generalizationeffects.

Of the 19 studies, three employed an alternating treatment design, comparing the effects of interventions using virtual manipulatives with interventions using either physicalmanipulativesornomanipulatives(Bouck,Chamberlain,& Park, 2017; Bouck, Shurr, et al., 2018; Satsangi et al., 2016). Nine studies examined the effects of interventionsusing virtual manipulatives by employing multiple-probe design across participants. Four studies used multiple-baseline design across participants (MBD-P). One study employed an MBD-P embedded alternating treatment design,which compared the effects of virtual manipulatives with acombined intervention using graphic organizers across participants (Hammer, 2018). The other two studies employed multiple-probe across behaviors design replicated acrossparticipants,amethodthatexaminestheeffectsofinterventions using virtual manipulatives across participants andmathematical domains (Bouck, Park, Cwiakala, Whorley,2020; Bouck, Park, Satsangi, et al., 2019).

Theparticipantsconsistedoffourelementary(11.4%),

16 middle (45.7%), and 15 high school-level students (42.9%). Most participants were male students (n=30) asopposed to female students (n=5). Of the 35 participants,24studentswereidentifiedashavingLDinmathematics

and the other five students were described as havingIndividualized Education Plan goals in mathematics. Theother six students with LD had not indicatedIndividualized

Education Program (IEP) goals in mathematics, but theyshowed low achievement on standardized mathematics assessments.Tobespecific,threeofthesesixstudentsscored below the 30th percentile on the mathematics problem-solving subtest of the Stanford AchievementTest(Xinet al., 2020), one scored in the 5th percentile on the problem-solving subtest of KeyMath (Reneau, 2013), one was a sixth-grade student who scored the equivalency of a 2.5-grade level (Park et al., 2021), and the other student scoredin the second percentile on the Northwest Evaluation Association standardized assessment (Simsek, 2016).

The mathematics domains of measurement were categorized according to CCSSM (NGA Center & CCSSO, 2010)asfollows:NumberandOperationsFractions(sixstudies),Operations and Algebraic Thinking (three studies), HighSchoolAlgebraReasoningwithEquationsandInequalities(three studies), Number and Operations in Base 10 (threestudies), Ratios and Proportional Relationships (one study), TheNumberSystem(onestudy),ExpressionsandEquations(one study), and Measurement and Data (one study).Regarding study quality,10 studies met all the criteria of CECindicators(Cooketal.,2014),receiving22points.Ninestudies missed one criterion, scoring 21 points. Twostudiesscored20points(seeonlinesupplementalTableS1).

Features of Mathematics Interventions Using Virtual Manipulatives

Inall19studies,studentsreceivedtheinterventionsinaone-on-one format. In three of the 19 studies (Shin & Bryant, 2017;Simsek,2016;Xinetal.,2020),technologyservedastheprimaryinterventionagentwithinatechnology-supported instruction environment (game and tutorials); researchersmonitored students' learning behaviors and prompted themwith expected steps, and students worked directly withpro-grams or apps without in-person instruction. In the other16studies, researchers delivered the interventions. Of the 16,one study described teacher-ledinstruction (Satsangi &Bouck,2015),and15studiesindicatedthatresearchers pro-videdexplicitinstructioninpretraining,instructionallessons, orinterventionphases.Inadditiontoexplicitinstruction,sixout of the 15 studies used a graduated instructional framework (i.e., CRA, VRA, virtual-representational [VR], orvirtual-abstract [VA]).Specifically,one study (Reneau,2013)connectedvirtualmanipulativestoaCRAinstructionalframework, two studies (Bouck, Bassette, et al., 2017; Parket al., 2021) examined the effectsof interventions using virtual manipulatives in accordance with the VRA framework, andtheotherthreestudiesreducedtheVRAtoeitheraVAor a VR framework. In all six studies using the graduated instructional framework, students were allowed to manipulate virtual manipulatives during the virtual phase only (seeonline supplemental TableS2).

Regardingthespecificuseofvirtualmanipulativesinthevariousstudies,sixstudiesimplementedinterventionsusingvirtualmanipulativeswithcomputertechnology,whereas13studies used touchscreen devices such as iPads. Most studiesusedapplicationsorprogramsdevelopedbyvendors.Forexample,10studiesutilizedvirtualmanipulativeappsdeveloped by Braining camp LLC, including Fraction Tiles,Base10 Blocks, Fraction Manipulatives, Cuisenaire Rods, Algebra Tiles,Color Tiles,and Two-ColorCounters. Fourother studies used virtual manipulative programs that weredeveloped by the National Library of Virtual Manipulatives(NLVM).In three studies, researchers developed their ownsoftware programs (Hammer, 2018; Shin & Bryant, 2017;Xin et al., 2020). One researcher-developed program (Shin& Bryant, 2017) included multi-components for teachingfractions (i.e., cognitive and metacognitive strategies,graphic organizers, and explicit instruction), another studyused an intelligent tutor-assisted program (Xin et al.,2020), and the third involved interventions involving technology-based graphic organizers and virtual manipulatives(Hammer, 2018). In terms of virtual manipulative types,three studies used single-representation virtual manipulatives, and seven studies used tutorial apps/programs. Onestudy indicated the use of two apps, single-representation and tutorial; eight studies indicated multi-representation; and only one study used virtual manipulatives embedded ingaming (see online supplemental TableS3).

Effects of Mathematics Intervention UsingVirtual Manipulatives

Table1 depicts the Tau-UES of immediate outcomes, maintenance, and generalization. From 19 studies, weextracted a total of 1,732 pairs, involving 35 participants,generating43Tau-UEStoexaminetheeffectsoftheimme- diate outcomes. The Tau-Ufor immediate outcomes indi-cated considerable effects from interventions using virtualmanipulatives in enhancing the mathematical performance of elementary and secondary school students with LD. Atthe AB phase level, the Tau-Uvaried from .56 to 1.00, butinthemajorityofstudies(95%,n=18),theTau-UESwere1.00,indicatingverylargeeffectsfrombaselinetointervention phases. Overall, interventions using virtual manipula- tives through explicit instruction, with a gradualsystematic approach such as VRA (e.g., Bouck, Bassette, et al., 2017)orwithoutthegradualshiftofrepresentation(e.g.,Hammer,2018), consistently showed large immediate outcomeeffects across the studies (Tau-U= 1.00). Furthermore,ininterventionswherestudentswithLDindependentlyusedvirtual manipulatives with the teacher-guided system of least prompts, following explicit training on use of virtualmanipulatives (e.g., Satsangi, Hammer, & Hogan, 2018),studentsdemonstratedlargeimprovementsfrombaselinetointervention phases (Tau-U=1.00). In three other studies(Shin & Bryant, 2017; Simsek, 2016; Xin et al., 2020) where students with LD received technology-assisted instruction with interventionists' prompts for instructionalsteps, students generally showed large immediate learning outcomes; students with LD exhibited very large effects on word problems involving multiplication and division ofwhole numbers (Xin et al., 2020) and moderate to largeintervention effects on learning the concepts of fractions (Shin & Bryant, 2017; Simsek, 2016). Although one of thethree middle school students with LD using a web-based computer application (Shin & Bryant, 2017) demonstratednoimprovementinthefirstinterventionsession,thestudent showed rapid growth (10.33% increase for every new session) during the intervention.

In addition, we examined whether students maintained the learned skills after the intervention. From 13 studies having maintenance effects data, we used a total of 30 BCphases involving 24 participants, and we calculated 662 pairs and 30 Tau-U.The individual Tau-Uof BC contrasts showed a wide range of outcomes from 1.0 to 0.67.Specifically, in eight (62%) out of 13 studies (Bouck, Bassette, et al., 2017; Bouck, Park, & Stenzel, 2020;Bouck & Park, 2020; Hammer, 2018; Park et al., 2021; Satsangi, Hammer,&Evmenova,2018;Satsangi,Hammer,&Hogan, 2018;Simsek,2016),participantsmaintainedtheirskillsgained from intervention at 100% (Tau-U =.00) or even showed higher improvement rates than in theirintervention phases (Tau-U=.00.41). Across the eight studies, themaintenance phase varied among conditions (i.e.,identical to baseline/best treatment/intervention) and periods (i.e.,14 weeks after the intervention). In three other studies (Bouck, Park, Cwiakala, & Whorley, 2020; Bouck, Park,Satsangi, et al., 2019; Bouck, Park, & Shurr, 2019), stu-dents'gains from the intervention were notmaintained (Tau-U=1.00to.47);inthesestudies,2weeksafterthe lastintervention(i.e.,explicitinstructionthroughVAorVR framework with the system of least prompts), maintenance sessions designed to be identical to the baseline condition (i.e., no prompts, manipulatives, or support) were conducted. In the two remaining studies (Satsangi & Bouck, 2015; Xin et al., 2020), two out of six students with LD showed slight improvements in the maintenance phase (Tau-U =.00 and .67), but the other four students withLD did not maintain their intervention effects using virtual manipulatives (Tau-U=.70 to .11). Todetermine generalization effects, we extracted atotal of 19 AD phases and 251 pairs involving 13 participantsfromsevenstudieshavinggeneralizationphasedata.Intwostudies (Bouck, Chamberlain, & Park, 2017; Bouck, Park, Levy,et al., 2020), students with LD demonstrated a low degreeofgeneralizationeffectscomparedwiththeirbaselinephases(Tau-U=.40to.20).Inthesestudies,the researchers designed the generalization sessions to beidenticaltobaselineconditions(i.e.,novirtualmanipulatives, teacher prompts, or supports). In the other five studies, studentswithLDdemonstratedahighdegreeofgeneralization effectscomparedwiththeirbaselinemathematicalperformances (Tau-U=1.00). Of these five studies, three (Bouck&Park,2020;Bouck,Shurr,etal.,2018;Hammer,2018)also conducted generalization sessions consistent with the baseline (i.e., no manipulatives, prompts, or supports), while the other two studies (Satsangi & Bouck, 2015; Satsangi, Hammer, & Evmenova, 2018) were designed to determine whether students could apply the skills they gained from the intervention to solve word problems or equations involving negative coefficients.

Discussion

Thepurposeofthisstudywastosynthesizetheliteratureoninterventions using virtual manipulatives to improve themathematics performance of K-12 students with LD. Wespecifically aimed to review the features of mathematics interventions using virtual manipulatives to support students with LD and examine the effects of mathematics interventionsusingvirtualmanipulativesonthemathematical achievement of students with LD. A total of 19 single-case design studies published between 2014 and 2020 werereviewed. As shown by the publication range, research onthe use of virtual manipulatives for students with LD has increased in recent years. All reviewed studies indicatedhighmethodologicalrigor,morethanhalfmetallcriteriaofCEC indicators (Cook et al., 2014), and the others missedonly one or two criteria. In the increasing interest and therigor of these studies, we have found some promising elements related to the efficacy of virtual manipulatives use in mathematics instruction for students with LD.

Features of Mathematics Interventions UsingVirtual Manipulatives

Fiveprograms(i.e.,AlgebraBalanceScale,AlgebraBalanceScaleforNegatives,FunFraction,Polynomials,andPGBM-COMPS) were used on a computer, and 10 programs (i.e.,Fraction Tiles,Base 10 Blocks, Fraction Manipulatives,Cuisenaire Rods, Algebra Tiles,Color Tiles,Two-ColorCounters, Conceptua Fractions, Thinking Blocks, andMotionMath)wereusedontouch-baseddevices(e.g.,iPads,tablets). The most commonly used types of virtualmanipulatives were multi-representation (n=8) and tutorial (n= 8, both of which allow participants to connect two ormore mathematical representations, including visual representation. Tutorial programs, in particular, provided features of guiding, tutoring, and instant feedback for the students.

Students connect different mathematical features andproperties and understand mathematical concepts throughanalysis and interaction with various representations(NCTM, 2014). The virtual manipulatives described in this synthesis support multi-representations in addition toconcrete representations (Moyer-Packenham & Bolyard, 2016). For example, Two-Color Counters and Cuisenaire Rods provide digital objects with labels for numbers or signs (e.g., positive, negative), providing concrete and symbolic representations. Other virtual manipulatives (e.g., Algebra Tiles) also show labeled tiles representing variables or constants. Technology-based manipulatives facilitate the provision of multiple mathematics representations. Given the built-in support of virtual manipulatives, students may experience a decreased cognitive load when using virtual manipulatives (Suh & Moyer-Packenham, 2008). For example, Fraction Tiles manipulatives do not allow 1, 7, 9, or 11 as denominatorsthus restricting students' choices and possibly reducing their cognitive loads (Bouck et al., 2017). Another virtual manipulatives application, Base 10 Blocks manipulatives allows students to regroup only the minuend; blocks for the subtrahend can-not be regrouped. In this same application, the selected blocks become semitransparent, which might support students' cognition during the problem-solving process (Bouck, Chamberlain, & Park, 2017). The Algebra Balance Scale (NLVM) separates positive and negative coefficients; therefore, students need to use Algebra Balance Scale for Negatives to work with equations having negative variables or constants. Despite the potential benefits of virtual manipulatives in terms of cognitive load, it should be noted that the studies reviewed in this synthesis lacked detailed

descriptions of the virtual manipulatives employed.

Effects of Mathematics Intervention UsingVirtual Manipulatives

The overall findings suggest that interventions usingvirtual manipulatives have positive effects on the mathematicalimprovement of students with LD across mathematical topics (e.g., whole number computations and the concept offractions). Fifteen studies that implemented interventionsusingvirtualmanipulativesthroughexplicitinstructionindicatedlargeimmediateoutcomeeffectswithoutregardtothegradual shift of representation (e.g., VRA) or the teacher-guided system of least prompts (e.g., Satsangi, Hammer, &Hogan, 2018). In the other three studies that provided technology-assisted instruction (Shin & Bryant, 2017; Simsek,2016;Xinetal.,2020),mostparticipantsalsoincreasedtheir outcomes with large intervention effects; however, oneparticipant showed a moderate immediate outcome effect.These promising outcomes are consistent with a previoussynthesis examining the effects of virtual manipulatives onmathematicsachievementamongthegeneralschoolpopulation (Moyer-Packenham & Westenskow,2013). Thesefindings indicate that the use of virtual manipulatives withinteacher-led or teacher-guided instruction and technology-assisted instruction can help students improve their mathematical performance. In particular, although students with LDdemonstrateawiderangeofimmediatelearningeffects,the use of technology-assisted instruction thatincludedfeaturesofvirtualmanipulativesdemonstratedapotentialforrapidgrowthduringtheintervention(Shin&Bryant,2017).ManystudentswithLDwhoreceivedmathematicsinterventionsusingvirtualmanipulativesmaintained(ineight outof13studieswithmaintenancedata)andgeneralized theinterventioneffects(infiveoutofsevenstudieswith generalization data). However, in some cases,althoughstudents with LD demonstrated very largeimmediatelearning effects, the same students had difficultyinmaintaining and/or generalizing those immediate learningeffects.In fivestudies,participantsdidnotfullymaintaintheskills obtained from the intervention; three of thesestudies(e.g., Bouck, Park, Cwiakala, & Whorley, 2020;Bouck,Park, Satsangi,etal.,2019;Bouck,Park,&Shurr,2019)employed explicitinstructionthroughareducedgradualsystematic approach(i.e.,VA,VR),andtheothertwostudiesdelivered eitherteacher-ledinstruction(e.g.,Satsangi&Bouck,2015) ortechnology-assistedinstruction(e.g.,Xinetal.,2020). ThisfindingindicatesthatsomestudentswithLDmight requiremorefrequentandintensivemathematicsinstruction. Wealso need further research toreachconclusions regardingtheeffectsofinterventionusingvirtualmanipulativesinthemaintenanceandgeneralizationphases.Ofthe 19 studies, six studies did not measuremaintenanceeffects and 12 studies did not include a generalization phase;threestudies included neither maintenance nor generalization.

Limitations and Future Research Directions

Although the overall findings suggest that interventionsusing virtual manipulatives are effective in improving mathematics achievement for students with LD, thisresearch has limitations that should be considered wheninterpreting the results. The primary concern is that therewere only 19 published studies on the use of virtualmanipulatives for students with LD. Tenof the 19 studies in this review included students with other disabilities, so weselectively coded for students with LD by disaggregating data for our target students. Thus, only 35 participantswereinvolved across 19 studies, limiting the generalizability ofour findings. Furthermore, we found no randomized controlled trial or quasi-experimental design studies that metour inclusion criteria. Future researchers should conduct morestudiesontheuseofvirtualmanipulativesandshould validate the intervention effects for students withLD.

Another limitation of this study is the difficulty ofclearly determining the unique features of virtual manipulatives during students' interactions with virtual manipulatives. Although virtual manipulatives model and imitate the shapes of physical manipulatives, virtual manipulatives also include features exclusive to technological tools. Some studies described the unique features of virtual manipulatives'apps,includinginherentbuilt-inconstraints.

For example, the small-sized screen of an iPad mightincrease students' errors in counting due to the difficulty ofseeing all the blocks (Bouck et al., 2017). However, most studies of virtual manipulatives provided little detail regarding how the students worked with virtual manipulativesduringtheirmathematicslearning(Lafayetal.,2019). To help readers understand the features of virtual manipulatives used in each study and to promote possible replication in the future, researchers should provide detailed descriptions of how students can manipulate the virtual manipulatives for their mathematicslearning.

The last limitation of this study relates to the noveltyeffect of using new technology. Although manyschool-age children use computers and touch-based devices, virtualmanipulatives are still an emerging technology in the educational field; research into mathematics interventionsusing virtual manipulatives is still under active review. Becausemostofthereviewedstudiesnotedthattheparticipants had no prior experience with virtual manipulatives, the novelty effect may have played a role in their findings. Thus, future researchers should consider the possibilitythatshort-term impacts may be due to the novelty of using virtual manipulatives in mathematics classes.

Implications for Practice

School-age children increasingly use personal devices (i.e., computers, iPads) and, as the use of technology hasbecome fundamental to the field of education, teachers may consider using computers or iPads as effective learning tools. Recent research findings indicate that students with LD favor the use of virtual manipulatives for their mathematics learning (Bouck et al., 2017; Satsangi et al., 2016). Virtual manipulatives require no storage space andno effort to assemble, enabling educators to access them easily and to plan differentiated instruction for the diverse learners in their classrooms (Bouck, Working, et al., 2018; Sayeski, 2008). In addition, educators may face difficulties in fitting specific physical manipulatives into their instruction, but virtual manipulatives can be used more flexibly (Moyer-Packenham et al., 2012; Satsangi et al., 2016). Furthermore, as an example of assistive technology, virtual manipulatives can support the mathematics learning of school-age students in online and blended learning environments (Serianni,2014).

This study suggests overall promising effects of inter-ventions using virtual manipulatives for the mathematics achievement of students with LD. Physical manipulativesare considered to be effective tools for teaching mathematics;however,educatorsareoftensensitivetothepossibility ofstigmatizingstudentswithLDinsecondaryandinclusive settings (Bouck, Bassette, et al., 2017; Satsangi & Bouck,2015; Satsangi et al., 2016). Virtual manipulatives, withtheirprovisionofmultiplerepresentations,areconsidered more socially acceptable tools for secondary students with LD. Thus, the use of virtual manipulatives can be one way to support students with LD in inclusive settings.